Applied Econometrics with

Time Series

Overview

Chapter 6

Time Series

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Overview

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Overview

Time series data: typical in macroeconomics and finance

Background reading: Brockwell and Davis (2002): Introduction to Time Series and Forecasting, 2nd edition.

Notation: yt , t = 1, . . . , n.

Brockwell and Davis (1991): Time Series – Theory and Methods, 2nd edition.

Contents: Infrastructure and “naive” methods ARMA modeling Stationarity, unit roots, and cointegration Time series regression and structural change Extensions (GARCH, structural time series models)

Franses (1998): Time Series Models for Business and Economic Forecasting Hamilton (1994): Time Series Analysis ...

Focus is on time domain methodology.

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Classes for time series data Standard time series class in R is “ts”: Aimed at regular series (annual, quarterly, monthly). A “ts” object is either a numeric vector (univariate series) or a numeric matrix (multivariate series).

Time Series

"tsp" attribute reflects time series properties:

Infrastructure and “Naive” Methods

a vector of length 3 with start, end and frequency. Create via ts(): supply data (numeric vector or matrix) plus arguments start, end, and frequency. Methods for standard generic functions: plot(), lines(), str(), summary(), . . . Additional time-series-specific methods: lag(), diff(), . . . .

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Classes for time series data 60000

Example: Quarterly consumption of non-durables in the United Kingdom (from Franses 1998) Plot: UKNonDurables

R> data("UKNonDurables") R> plot(UKNonDurables)

Time series properties: R> tsp(UKNonDurables) 4

30000

[1] 1955 1989

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50000

Classes for time series data

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40000

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Subsets via window(): R> window(UKNonDurables, end = c(1956, 4))

1955

Qtr1 Qtr2 Qtr3 Qtr4 1955 24030 25620 26209 27167 1956 24620 25972 26285 27659 Christian Kleiber, Achim Zeileis © 2008–2015

1960

1965

1970

1975

1980

1985

1990

Time

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Classes for time series data

(Linear) filtering Linear filter: important class are finite moving averages

Drawbacks of “ts”: Only numeric time stamps (more general date/time classes?) Missing values cannot be omitted (start/end/frequency no longer sufficient for reconstructing all time stamps!) – a problem with irregular series, e.g., with many financial time series. R packages for irregular series: several, we use zoo Generalization of “ts”: time stamps of arbitrary type. Numeric vectors or matrices, "index" attribute contains vector of time stamps (not just "tsp" attribute!). Regular series can be coerced back and forth between “ts” and “zoo” via as.zoo() and as.ts(). “zoo” more convenient for daily data (e.g., “Date” time stamps) or intraday data (e.g., “POSIXct” or “chron” time stamps). More details: Zeileis and Grothendieck (JSS 2005). Christian Kleiber, Achim Zeileis © 2008–2015

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(Linear) filtering

yˆt =

s X

aj yt +j ,

j =−r

t = r + 1, . . . , n − s.

If r = s, filter is called symmetric. In R: function filter() Main argument filter takes vector containing aj s. Can also apply recursive linear filters. Example: (UKDriverDeaths, Harvey and Durbin, JRSS A 1986) R> data("UKDriverDeaths") R> plot(UKDriverDeaths) R> lines(filter(UKDriverDeaths, c(1/2, rep(1, 11), 1/2)/12), + col = 2) Christian Kleiber, Achim Zeileis © 2008–2015

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(Linear) filtering

2500

Further examples:

rollapply() computes functions on moving data windows:

2000

filter() also provides autoregressive (recursive) filtering. Generate 100 observations from AR(1) process:

1500

UKDriverDeaths

R> plot(rollapply(UKDriverDeaths, 12, sd))

1000

R> set.seed(1234) R> x dd_dec dd_stl plot(dd_dec$trend, ylab = "trend") R> lines(dd_stl$time.series[,"trend"], lty = 2, lwd = 2) Applied Econometrics with R – 6 – Time Series – 12 / 67

Decomposition

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Decomposition 7.8 7.0

7.4

data

7.8 7.4

1970

1975

1980

1985

−0.15

0.00

remainder

0.15

7.2

7.4

trend

7.6

−0.1

0.1

seasonal

7.67.0 7.4 0.1 0.15−0.1 0.00

random

−0.15

seasonal

7.2

trend

observed

Decomposition of additive time series

1970

Time Christian Kleiber, Achim Zeileis © 2008–2015

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1980

1985

time Applied Econometrics with R – 6 – Time Series – 14 / 67

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Decomposition

Exponential smoothing

7.6

HoltWinters() handles exponential smoothing and generalizations: Recursively reweighted lagged observations for predictions.

7.4

Default: Holt-Winters filter with additive seasonal component. Example: UKDriverDeaths

7.3

trend

7.5

Smoothing parameters determined by minimizing squared prediction error on observed data.

Historical sample up to 1982(12) (before change in legislation). 7.2

Use Holt-Winters to predict observations for 1983 and 1984.

1970

1975

1980

1985

R> plot(dd_hw, dd_pred, ylim = range(UKDriverDeaths)) R> lines(UKDriverDeaths)

Time

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R> dd_past dd_hw dd_pred ar(x) Call: ar(x = x)

Order selected 1

0.4 0.2

Fit autoregression to x via ar():

ACF

set.seed(1234) x R> R> R>

Series x

1.0

In R: acf() and pacf().

15

20

5

10

15

20

Lag

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Classical model-based analysis

Classical model-based analysis

0

1

seasonal and nonseasonal differences

3

4

1

Classical model-based analysis

0.5

Partial ACF

0.5

−1.0 1

2

3

4

Christian Kleiber, Achim Zeileis © 2008–2015

1

2

3

4

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Classical model-based analysis

Preliminary analysis suggests double differencing (d = 1, D = 1), some AR and MA effects – we use p = 0, 1, 2 and q = 0, 1, 2, low-order seasonal AR and MA parts – we use P = 0, 1 and Q = 0, 1. This gives 36 parameter combinations in total. Manual solution: Set up all parameter combinations via expand.grid(). Fit each SARIMA model using arima() in for() loop. Store resulting BIC extracted from the model. For BIC, use AIC() with k = log(length(nd)).

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4

1.0

Series diff(diff(nd, 4))

1.0

Series diff(diff(nd, 4))

Lag Applied Econometrics with R – 6 – Time Series – 24 / 67

3 Lag

0.0 0

Christian Kleiber, Achim Zeileis © 2008–2015

2

Lag

−1.0

acf(diff(nd), ylim = c(-1, 1)) pacf(diff(nd), ylim = c(-1, 1)) acf(diff(diff(nd, 4)), ylim = c(-1, 1)) pacf(diff(diff(nd, 4)), ylim = c(-1, 1))

ACF

R> R> R> R>

2

0.0

nonseasonal differences

−1.0

−1.0

Empirical ACFs and PACFs for

0.5

Partial ACF

0.0

ACF

0.5

R> nd + R> R> + + R>

nd_pars plot(log(UKNonDurables)) R> lines(nd_pred$pred, col = 2)

0.0 0.6

p value

R> nd_pred R> R> +

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Unit-root tests Available tests:

black white

5000

Augmented Dickey-Fuller (ADF) test: t test of H0 : % = 0 in

∆yt = α + δ t + %yt −1 +

3000

In R: adf.test() from tseries.

k X j =1

φj ∆yt −j + εt .

Phillips-Perron (PP) test: Same idea as ADF, but nonparametric (HAC) correction for autocorrelation. In R: pp.test() from tseries.

1000

PepperPrice

7000

Stationarity, unit roots, and cointegration

data("PepperPrice") plot(PepperPrice, plot.type = "single", col = 1:2) legend("topleft", c("black", "white"), bty = "n", col = 1:2, lty = rep(1,2))

1975

1980

1985

1990

1995

Time

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Elliott-Rothenberg-Stock (ERS): Same idea as ADF, but GLS detrending. In R: ur.ers() from urca.

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Unit-root tests

Unit-root tests

ADF in levels:

PP in levels (by default with time trend):

R> library("tseries") R> adf.test(log(PepperPrice[, "white"]))

R> pp.test(log(PepperPrice[, "white"]), type = "Z(t_alpha)")

Augmented Dickey-Fuller Test data: log(PepperPrice[, "white"]) Dickey-Fuller = -1.7, Lag order = 6, p-value = 0.7 alternative hypothesis: stationary

Phillips-Perron Unit Root Test data: log(PepperPrice[, "white"]) Dickey-Fuller Z(t_alpha) = -1.6, Truncation lag parameter = 5, p-value = 0.7 alternative hypothesis: stationary

ADF in first differences: R> adf.test(diff(log(PepperPrice[, "white"]))) Augmented Dickey-Fuller Test data: diff(log(PepperPrice[, "white"])) Dickey-Fuller = -5.3, Lag order = 6, p-value = 0.01 alternative hypothesis: stationary Warning message: In adf.test(diff(log(PepperPrice[, "white"]))) : p-value smaller than printed p-value Christian Kleiber, Achim Zeileis © 2008–2015

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Stationarity tests

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Stationarity tests

Kwiatkowski, Phillips, Schmidt and Shin (J. Econometrics 1992):

KPSS without time trend:

Test H0 :

R> kpss.test(log(PepperPrice[, "white"]))

rt ≡ 0 in

KPSS Test for Level Stationarity

yt = dt + rt + εt , where dt deterministic trend,

data: log(PepperPrice[, "white"]) KPSS Level = 0.91, Truncation lag parameter = 3, p-value = 0.01 Warning message: p-value smaller than printed p-value in: kpss.test(log(PepperPrice[, "white"]))

rt random walk,

εt stationary (I (0)) error process. Two variants: dt = α, level stationarity (under H0 ). dt = α + β t, trend stationarity (under H0 ).

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Cointegration

Cointegration

Pepper series exhibit common nonstationary features.

R> po.test(log(PepperPrice))

Cointegration tests in R:

Phillips-Ouliaris Cointegration Test

Engle-Granger two-step method Available in po.test() from tseries (named after Phillips and Ouliaris, Econometrica 1990). Johansen test Full-information maximum likelihood approach in pth-order cointegrated VAR. Error correction form (ECM) is (without deterministic components)

∆yt = Πyt −1 +

Engle-Granger two-step with black pepper regressed on white pepper:

p−1 X j =1

data: log(PepperPrice) Phillips-Ouliaris demeaned = -24, Truncation lag parameter = 2, p-value = 0.02

Suggests both series are cointegrated. Remarks: Test with reverse regression is

po.test(log(PepperPrice[,2:1])) Γj ∆yt −j + εt .

Problem: treatment asymmetric, but concept cointegration demands symmetric treatment!

Trace and lambda-max tests available in ca.jo() from urca. Christian Kleiber, Achim Zeileis © 2008–2015

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Cointegration

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Cointegration

Johansen test with constant term

Values of teststatistic and critical values of test:

R> library("urca") R> pepper_jo summary(pepper_jo)

test 10pct 5pct 1pct r dd dd_dat lm(dd ~ dd1 + dd12, data = dd_dat) Call: lm(formula = dd ~ dd1 + dd12, data = dd_dat) Coefficients: (Intercept) 0.421

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More on fitting dynamic regression models Approach 2: use convenience interface dynlm() from dynlm

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Structural change tests Features of UKDriverDeaths: Parameters of time series model unlikely to be stable throughout sample period.

Time series regression with "ts" data: Start = 1970(1), End = 1984(12)

Package strucchange implements large collection of tests for structural change (parameter instability).

Call: dynlm(formula = dd ~ L(dd) + L(dd, 12)) L(dd) 0.431

Christian Kleiber, Achim Zeileis © 2008–2015

dd12 0.511

Decrease in mean number of casualties after policy change.

R> library("dynlm") R> dynlm(dd ~ L(dd) + L(dd, 12))

Coefficients: (Intercept) 0.421

dd1 0.431

Two types of tests:

L(dd, 12) 0.511

Fluctuation tests. Tests based on F statistics.

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Structural change tests

Structural change tests

Fluctuation tests:

Fluctuation tests in strucchange:

Assess structural stability by capturing fluctuation in CUSUMs or MOSUMs of

empirical fluctuation processes via efp(). Result is object of class “efp”.

residuals (OLS or recursive), model scores (empirical estimating functions), or parameter estimates (recursive or rolling).

Idea: under null hypothesis of parameter stability, resulting “fluctuation processes” exhibit limited fluctuation, under alternative of structural change, fluctuation is generally increased. Evidence for structural change if empirical fluctuation process crosses boundary that corresponding limiting process crosses only with probability α.

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Structural change tests

plot() method for performing test graphically. sctest() method (for structural change test) for traditional significance test. Example: OLS-CUSUM for UKDriverDeaths OLS-CUSUM process: Scaled CUSUM of OLS residuals εˆt = yt − xt> βˆ efp(s) =

1

√ σ ˆ n

bnsc X t =1

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εˆt ,

0 ≤ s ≤ 1.

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Structural change tests OLS−based CUSUM test

data: dd_ocus S0 = 1.5, p-value = 0.02 R> plot(dd_ocus)

1.0 0.5 0.0

OLS-based CUSUM test

−0.5

R> sctest(dd_ocus)

−1.0

Test using maximum absolute deviation of efp (default functional)

Empirical fluctuation process

R> library("strucchange") R> dd_ocus dd_fs sctest(dd_fs) supF test data: dd_fs sup.F = 19, p-value = 0.007

Visualization: R> plot(dd_fs, main = "supF test")

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Structural change tests

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Structural change tests supF test

20

Further Example: German M1 money demand Lütkepohl, Teräsvirta and Wolters (JAE 1999) use error correction model (ECM) for German M1.

15

GermanM1 contains data from 1961(1) to 1995(4) on per capita

10

Load and set up model R> data("GermanM1") R> LTW m1_re plot(m1_re) Christian Kleiber, Achim Zeileis © 2008–2015

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Structural change tests

Dating structural changes

RE test (recursive estimates test)

Setup is linear regression model t = nj −1 + 1, . . . , nj ,

j = 1, . . . , m + 1,

1.5

where

0.5

1.0

j = 1, . . . , m segment index, β (j ) segment-specific set of regression coefficients, {n1 , . . . , nm } set of unknown breakpoints (convention: n0 = 0 and nm+1 = n). In R: function breakpoints()

0.0

1985

1990

1995

Time Applied Econometrics with R – 6 – Time Series – 56 / 67

Dating structural changes

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Dating structural changes

Example: UKDriverDeaths

●

(Intercept) dd1 dd12 1970(1) - 1973(10) 1.458 0.1173 0.6945 1973(11) - 1983(1) 1.534 0.2182 0.5723 1983(2) - 1984(12) 1.687 0.5486 0.2142

−220 −240

R> coef(dd_bp, breaks = 2)

● ● ●

−260

R> dd_bp plot(dd_bp, legend = FALSE, main = "") R> plot(dd) R> lines(fitted(dd_bp, breaks = 2), col = 4) R> lines(confint(dd_bp, breaks = 2))

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−300

●

Visualization

● ●

●

0

●

●

2

●

4

6

●

●

1.2

Empirical fluctuation process

2.0

yt = xt> β (j ) + εt ,

8

Number of breakpoints

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7.6

Time Series

7.4

Extensions

7.0

7.2

dd

7.8

Dating structural changes

1970

1975

1980

1985

Time

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Extensions

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Structural time series models

Further packages for time series analysis dse – Multivariate time series modeling with state-space and vector ARMA (VARMA) models. FinTS – R companion to Tsay (2005). forecast – Univariate time series forecasting, including exponential smoothing, state space, and ARIMA models. fracdiff – ML estimation of ARFIMA models and semiparametric estimation of the fractional differencing parameter. longmemo – Convenience functions for long-memory models. mFilter – Time series filters, including Baxter-King, Butterworth, and Hodrick-Prescott. Rmetrics – Some 20 packages for financial engineering and computational finance, including GARCH modeling in fGarch. tsDyn – Nonlinear time series models: STAR, ESTAR, LSTAR. vars – (Structural) vector autoregressive (VAR) models Christian Kleiber, Achim Zeileis © 2008–2015

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Basic structural model has measurement equation yt = µt + γt + εt ,

εt ∼ N (0, σε2 ) i.i.d.

Seasonal component γt (with frequency s) is

γt + 1 = −

s −1 X j =1

γt +1−j + ωt ,

ωt ∼ N (0, σω2 ) i.i.d.

Local level and trend components are

µt +1 = µt + ηt + ξt , η t + 1 = η t + ζt ,

ξt ∼ N (0, σξ2 ) i.i.d.,

ζt ∼ N (0, σζ2 ) i.i.d.

All error terms mutually independent. In R: R> dd_struct plot(cbind(fitted(dd_struct), residuals(dd_struct))) Christian Kleiber, Achim Zeileis © 2008–2015

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Structural time series models

GARCH models 3

7.7 7.5

2

7.3

1 0

MarkPound

0.002 7.1 −0.002 −0.006 0.2

−1

0.1 0.0

−2

3 −0.2 −0.1 2 1 0

0

500

−3 −2 −1

residuals(dd_struct) fitted(dd_struct).sea fitted(dd_struct).slope fitted(dd_struct).level

cbind(fitted(dd_struct), residuals(dd_struct))

1000

1500

2000

Time 1970

1975

1980

1985

Time

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GARCH models

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GARCH models

tseries function garch() fits GARCH(p, q) with Gaussian innovations. Default is GARCH(1, 1): yt

= σt νt ,

σt2

νt ∼ N (0, 1) i.i.d.,

= ω + αyt2−1 + βσt2−1 ,

ω > 0, α > 0, β ≥ 0.

Example: DEM/GBP FX returns for 1984-01-03 through 1991-12-31

Coefficient(s): Estimate Std. Error a0 0.0109 0.0013 a1 0.1546 0.0139 b1 0.8044 0.0160

t value Pr(>|t|) 8.38